It has to be noticed that the active nature of has been
completely ignored in the above arguments:
the prediction for the structure function given by Eq. (2.31)
is identical to Eq. (2.29) obtained for the passive
scalar with finite life-time.
The crucial hypothesis in the derivation of Eq. (2.31) is
that we have assumed that the statistics of trajectories is
independent of the forcing
:
The random variable arises from forcing contributions
along the trajectories at times
, when the distance
between the two fluid particles is larger than the
forcing correlation length
, whereas the
exit-time
is clearly determined by the evolution of the strain
at times
.
Since the correlation time of the strain is
, for
We remark that, were the velocity field non-smooth, the exit-times would be
independent of in the limit
and the above argument
would not be relevant. Therefore, the smoothness of
the velocity field plays a central role in the equivalence of
vorticity and passive scalar statistics.